Ya. It should be chainable. And they are just syntactic sugar for STL fan like me.

The meaning is up to Qt developer to decide. They can be anything. Different module can bring different meaning. Not a problem. Example from SOCI, database library:-
@
session sql;
sql.open(postgresql, "dbname=mydb");
sql << "drop table person";
sql << "select id, name from company";

In all real-world examples you've mentioned, those are stream-like structures. And those do have << and >> operators defined in Qt.

The problem (from my point of view) with having the operators in question do fancy stuff like you're proposing is that in STL they are not fancy at all. For example, it would be like introducing into std::map something like this: